Longer, deeper and affordable kisses. · Folding kissing hairpins Stefan Janssen (Bielefeld University) Longer, deeper and a ordable kisses. LIX 2010, Paris 13 / 28 12 moving boundaries!O(n12)

Longer, deeper and affordable kisses.

Corinna Theis, Stefan Janssen and Robert Giegerich

Faculty of Technology & Center for BiotechnologyBielefeld University, Germany

[email protected]

LIX Bioinformatics Colloquium,8-10th November 2010,

Ecole Polytechnique, Paris

Stefan Janssen (Bielefeld University) Longer, deeper and affordable kisses. LIX 2010, Paris 1 / 28

Folding nested structures


maximizing base-pairs

minimizing free energy

O(n3) time

O(n2) space

Folding crossed structures


general pseudoknotfolding is NP-hard[Lyngsø 2000, Akutsu 2000]

polynomial algorithmsfor special classes

Folding simple pseudoknots


Folding simple recursive pseudoknots


loop regions may contain substructures ...

... even simple recursive pseudoknots

already computed

Folding simple recursive pseudoknots


8 moving boundaries

→ O(n8) time→ O(n2) space

too much computing time

→ canonization= pknotsRG

[Reeder & Giegerich 2004]

Folding canonical simple recursive pseudoknots


Three rules of canonization1 perfect helices: no loops, no bulges




2 maximal helices




2 maximal helices

3 helices must not overlap

The clue

optimal helices can beprecomputed in O(n2) time andspace

pknotsRG: 4 moving boundaries


What canonization might miss


Folding kissing hairpins




12 moving boundaries


way too high compute time



12 moving boundaries


way too high compute time

with canonization


Kissing hairpin as an overlay of two pseudoknots


Problem 1: Violating Bellman’s Principle


Bellman’s Principle: Monotonicity of scoring wrto optimizationE (Kiss) = E (left PK ) + E (right PK )− E (overlap)

Problem 2: Incompatible pseudoknots


Strategy A


Summary Strategy A

Strategy A is correct, symmetric and heuristic:

1 Assume left PK optimal and find best consistent right PK

2 Assume right PK optimal and find best consistent left PK

3 Return better overlay structure


Strategy B


Strategy C


l = boundaryleft (rpk (k))rpk(k) = min (E (PKh,k+2,m,j) , rpk(k + 1))

Relating strategies

Strategies A,B,C are heuristic, but increasingly general.For any sequenxe x :

SearchspaceA(x) ⊆ SearchspaceB(x) ⊆ SearchspaceC (x) ⊂ SearchspacecsrKH(x)


O(n4) O(n4) O(n5) O(n6) timeO(n2) O(n3) O(n2) O(n2) space

2.4% 5.5% 2.4% 0% missed opt.csrKH

1 1.2 11.2 508.0 time1 9.2 1.3 4.5 space

Evaluation

Evaluation in terms of Pseudobase:61 pseudoknotted structures, including (only) 6 kissing hairpin motifsAgreement of strategies:

C 6= B 6= A C = B 6= A C = B = A

2 2 57

Correctness of predictions:

A,B,C agree and correctly predict the 6 true kissing hairpins

for 7 simpler pseudoknots, a kissing hairpin is predicted


Co-transcriptional folding


Strategy A: in all cases, it is the left PK which is locally optimal

→ double speed of Strategy A, by dropping symmetric case?

→ more complex motifs as successive overlays in O(n4) / O(n2)?

Co-transcriptional folding


Strategy A: in all cases, it is the left PK which is locally optimal

→ double speed of Strategy A, by dropping symmetric case?

→ more complex motifs as successive overlays in O(n4) / O(n2)?



Thanks for your attention

Problem with Strategy B


Extra Heuristic for Strategy B


Documents

Longer, deeper and affordable kisses. · Folding kissing hairpins Stefan Janssen (Bielefeld University) Longer, deeper and a ordable kisses. LIX 2010, Paris 13 / 28 12 moving boundaries!O(n12)